Problem: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $27$ years; the standard deviation is $2.6$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living longer than $24.4$ years.
Answer: $27$ $24.4$ $29.6$ $21.8$ $32.2$ $19.2$ $34.8$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $27$ years. We know the standard deviation is $2.6$ years, so one standard deviation below the mean is $24.4$ years and one standard deviation above the mean is $29.6$ years. Two standard deviations below the mean is $21.8$ years and two standard deviations above the mean is $32.2$ years. Three standard deviations below the mean is $19.2$ years and three standard deviations above the mean is $34.8$ years. We are interested in the probability of a snake living longer than $24.4$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the snakes will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the snakes will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $24.4$ years and the other half $({16\%})$ will live longer than $29.6$ years. The probability of a particular snake living longer than $24.4$ years is ${68\%} + {16\%}$, or $84\%$.